Using concept maps and interpretive essays for assessment in mathematics

School Science and Mathematics, Jan 1999 by Bolte, Linda A

This study explored the use of student-constructed concept maps in conjunction with written interpretive essays as an additional method of assessment in three undergraduate mathematics courses. The primary objectives of this study were to evaluate the benefits of using concept maps and written essays to assess the "connectedness" of students' knowledge; to measure the correlation between students ' scores on the concept maps and written essays, course exams, and final grade; and to document students ' perception of the effect of this approach on their mathematical knowledge. Results indicated that concept maps, when combined with written essays, are viable tools for assessing students' organization of mathematical knowledge. In addition, students perceive this approach as enhancing their mathematical knowledge.

Two aspects of a meaningful mathematical experience are strongly emphasized in the current literature: the development of an integrated knowledge-base in mathematics and the communication of mathematical knowledge (National Council of Teachers of Mathematics [NCTM], 1989; Roberts, 1996; Steen et al., 1990). The integration of the components into the mathematics curriculum necessitates (a) encouraging students to investigate the connections and interplay among various mathematical topics, so they can recognize equivalent representations of the same concept and (b) providing students with the opportunity to reflect on and clarify their own thinking about mathematical ideas and situations and to express mathematical ideas in writing.

Although widely accepted as worthwhile, this notion is largely based on intuition (Skemp, 1978). Direct measurement of students' organization of mathematical knowledge, the effect of this organization on their learning, and the use of concept maps as an assessment instrument in mathematics are relatively unexplored areas of research. This study investigated the combined use of concept maps and interpretive essays as an additional method of assessment in three undergraduate mathematics courses. The primary objectives of this study were to (a) evaluate the benefits of using concept maps and written essays to assess the connectedness of students' knowledge; (b) measure the correlation between students' scores on the concept maps and written essays, course exams, and final grade; and (c) document students' perceptions of the value of this approach. The associated quantitative and qualitative results indicated that concept maps, when combined with written essays, are viable tools for assessing students' organization of mathematical knowledge and that students perceive this approach as enhancing their mathematical knowledge.

Background

The use of concept maps as an indication of the connectedness of knowledge is largely based on the work of Novak (1984) in the area of science education. As graphical representations linking related concepts to form chains of relationships, concept maps have been "developed specifically to tap into a learner's cognitive structure and to externalize, for both the learner and the teacher to see, what the learner already knows" (Novak, 1984, p. 40). In a concept map, related concepts are represented as nodes, and the specific relationship between two concepts is indicated by linking words that are written along the line connecting the nodes (see Appendix A for student maps).

The use of concept maps to assess the organization of mathematical knowledge (Hasemann & Mansfield, 1995) is closely tied to a model developed by Hiebert and Carpenter (1992) for analyzing the issues of learning and teaching mathematics with understanding. This model, which flows from studies on knowledge structures in cognitive science (Chi, 1978; Quillian, 1968), is based on the assumption that internal representations of knowledge are connected in some useful way. These networks of knowledge are generally described metaphorically in two ways: as vertical hierarchies in which some representations subsume other representations and as web-like arrangements in which pieces of information and corresponding relationships between the items of information form simple linear chains or complex networks.

Educational applications of concept mapping include their use as a learning/study strategy and as an assessment instrument in a variety of settings (Bartels, 1995; Beyerbach, 1988; Mansfield & Happs, 1991; Novak, 1991). As noted by Angelo and Cross (1993), concept maps "stimulate students to create, and allow faculty to assess, original intellectual products that result from a synthesis of the course content and the students' intelligence, judgment, knowledge, and skills" (p. 181). Concept maps have also been used effectively within an instructional setting to identify student misconceptions(Bartels,1995; Kounba,1994; Novak, 1984).

The use of writing as a means of enhancing and assessing student understanding is consistent with the current trend to incorporate writing in mathematics classes (Connolly & Vilardi, 1989; Sterrett, 1990). Although the number of proponents of writing is substantial, research which verifies the benefits of writing as a means to enhance and assess mathematical understanding is somewhat limited and focuses primarily on the use of journals or writing prompts. (Borasi & Rose, 1989; Clarke, 1993; Miller, 1992; Miller & England,1989; Powell & Lopez,1989; Swinson,1992). Of particular interest is the use of samples of students' writing to identify misconceptions (Birken, 1989; Miller, 1992).